Astm stp 677 1979

SEEGER 144 Effect of the Active Plastic Zone on Fatigue Crack Growth Rates— GUNTER MARCI 168 A Comparative Experimental Study on the Fatigue Crack Closure Behavior Under Cyclic Loadi

FRACTURE MECHANICS

Proceedings of the

Eleventh National Symposium

on Fracture Mechanics: Part I

A symposium sponsored by

ASTM Committee E-24 on

Fracture Testing of Metals

AMERICAN SOCIETY FOR

TESTING AND MATERIALS

Virginia Polytechnic Institute

and State University

Blacksburg, Va., 12-14 June 1978

ASTM SPECIAL TECHNICAL PUBLICATION 677

C W Smith, Virginia Polytechnic

Institute and State University,

editor

List price $60.00

04-677000-30

#

(AMERICAN SOCIETY FOR TESTING AND MATERIALS

1916 Race Street, Philadelphia, Pa 19103

Copyright© AMERICAN SOCIETY FOR TESTING AND MATERIALS 1979

Library of Congress Catalog Card Number: 78-74567

NOTE The Society is not responsible, as a body, for the statements and opinions advanced in this publication

Printed in Baltimore, Md

August 1979

Foreword

This publication, Fracture Mechanics, contains papers presented at the

Eleventh National Symposium on Fracture Mechanics which was held 12-14 June 1978 at Virginia Polytechnic Institute and State University, Blacksburg, Va The American Society for Testing and Materials' Com-mittee E-24 on Fracture Testing of Metals sponsored the symposium C

W Smith, Virginia Polytechnic Institute and State University, served as editor of this publication

The proceedings have been divided into two volumes: Part \—fracture Mechanics and Part II—Fracture Mechanics Applied to Brittle Materials

Related ASTM Publications

Developments in Fracture Mechanics Test Methods Standardization, STP

A Note of Appreciation

to Reviewers

This publication is made possible by the authors and, also, the unheralded efforts of the reviewers This body of technical experts whose dedication, sacrifice of time and effort, and collective wisdom in review-ing the papers must be acknowledged The quality level of ASTM publications is a direct function of their respected opinions On behalf of ASTM we acknowledge with appreciation their contribution

ASTM Committee on Publications

Editorial Staff

Jane B Wheeler, Managing Editor Helen M Hoersch, Associate Editor Ellen J McGlinchey, Senior Assistant Editor

Helen Mahy, Assistant Editor

Contents

Introduction I

FATIGUE CRACK GROWTH STUDIES

Effect of Biaxial Stresses on Crack Growth—A F LIU, J E ALLISON,

D F DITTMER, AND J R YAMANE 5

Fatigue Crack Growtli Threshold in Mild Steel Under Combined Loading—

L P POOK AND A F GREENAN 23

Sequence Effects on Fatigue Crack Propagation; Mechanical and

Micro-structural Contributions—H NOWACK, K H TRAUTMANN, K

SCHULTE, AND G LUTJERING 36

Variations in Crack Growth Rate Behavior—M E ARTLEY, J P

GALLAGHER, AND H D STALNAKER 54

Application of Fracture Mechanics to Damage Accumulation in High

Temperature Fatigue—M J DOUGLAS AND A PLUMTREE 68

Cryogenic Effects on the Fracture Mechanics Parameters of Ferritic Nickel

Alloy Steels—R L TOBLER, R P MIKESELL, A N D R P REED 85

Evaluation of Temperature Effects on Crack Growth in Aluminum Sheet

Material—D E PETTIT A N D J M VAN ORDEN 106

Effects of Temperature and Frequency on the Fatigue Crack Growth Rate

Properties of a 1950 Vintage CrMoV Rotor Material—T T

SHIH AND G A CLARKE 125

Structural Memory of Cracked Components Under Irregular Loading—

H FURRING AND T SEEGER 144

Effect of the Active Plastic Zone on Fatigue Crack Growth Rates—

GUNTER MARCI 168

A Comparative Experimental Study on the Fatigue Crack Closure Behavior

Under Cyclic Loading for Steels and Aluminum Alloys—j A

VAZQUEZ, AUGUSTO MORRONE, AND J C GASCO 187

Effect of Residual Stresses on Fatigue Crack Growth in Steel Weldments

Under Constant and Variable Amplitude Loads—GRZECORZ GLINKA 198

Role of Crack-Tip Stress Relaxation in Fatigue Crack Growth—A SAXENA

AND S J HUDAK, JR 215

Crack Closure During Fatigue Crack Propagation—w j D SHAW AND

Fatigue at Notches and the Local Strain and Fracture Mechanics Approaches—

N E DOWLING 247

A Strain Based Intensity Factor Solution for Short Fatigue Cracks Initiating

from Notches—M H EL HADDAD, K N SMITH, A N D T H TOPPER 274

Cracli Initiation in a High-Strength Low-Alloy Steel—B L BRAGLIA, R W

HERTZBERG, AND RICHARD ROBERTS 290

Effect of Spherical Discontinuities on Fatigue Crack Growth Rate

Per-formance—W G CLARK, JR 303

Prediction of Fatigue Crack Growth Under Spectrum Loads—A E GEMMA

AND D, W SNOW 320

SURFACE FLAWS

Semi-Elliptical Cracks in a Cylinder Subjected to Stress Gradients—j

HELIOT, R C L A B B E N S , AND A PELLISSIER-TANON 341

Stress Intensity Factor Solutions for Internal Longitudinal Semi-Elliptical

Surface Flaws in a Cylinder Under Arbitrary Loadings—J j

MCGOWAN AND M RAYMUND 365

Theoretical and Experimental Analysis of Semi-Elliptical Surface Cracks

Subject to Thermal Shock—G YAGAWA, M ICHIMIYA, AND Y ANDO 381

Growth of Part-Through Cracks—L HODULAK, H KORDISCH, S

KUNZEL-MANN, AND E SOMMER 399

Stress Intensity Factors for Two Symmetric Corner Cracks—i s RAJU AND

J C NEWMAN, JR 411

Influence of Flaw Geometries on Hole-Crack Stress Intensities—c w

SMITH, W H PETERS, AND S F GOU 431

EXPERIMENTAL FRACTURE MECHANICS—X^ic, J,c,SPECIMEN GEOMETRY EFFECTS,

AND E X P E R I M E N T A L T E C H N I Q U E S

Variation of Fracture Toughness with Specimen Geometry and Loading

Conditions in Welded Low Alloy Steels—A PENELON, M N BASSIM,

AND J M DORLOT 449

7,t Results and Methods with Bend Specimens—J H UNDERWOOD 463

Investigation of Specimen Geometry Modifications to Determine the

Con-servative, JfR Curve Tearing Modulus Using the HY-130 Steel

System—J P GUDAS, J A JOYCE, AND D A DA VIS 474

An Experimental Study of the Crack Length/Specimen Width (a/W) Ratio

Dependence of the Crack Opening Displacement (COD) Test Using

Small-Scale Specimens—P M S T DE CASTRO, J SPURRIER, A N D

Dynamic Photoelastic and Dynamic Finite Element Analyses of Polycarbonate

Dynamic Tear Test Specimens—s MALL, A S KOBAYASHI, AND Y

URABE 498 Effect of Specimen Geometry on Crack Growth Resistance—s j GARWOOD 511

Single-Edge-Cracked Crack Growth Gage—j A ORI AND A F GRANDT, JR 533

Measurement of Crack-Tip Stress Distributions by X-Ray Diffraction—J E

ALLISON 550

Correlations Between Ultrasonic and Fracture Toughness Factors in Metallic

Materials—ALEX VARY 563

SPECIAL TOPICS

Analysis of Load-Displacement Relationships to Determine J-R Curve

and Tearing Instability Material Properties—HUGO ERNST,

p C PARIS, MARK ROSSOW, AND J W HUTCHINSON 5 8 ]

Path Dependence of J in Three Numerical Examples—M E

KARABIN, JR., AND J L SWEDLOW 600

Description of Stable and Unstable Crack Growth in the Elastic

Plastic Regime in Terms of/r Resistance Curves—c E TURNER 614

Strain Energy Release Rate Method for Predicting Failure Modes in

Com-posite Materials—R s WILLIAMS AND K L REIFSNIDER 629

An Analysis of Tapered Double-Cantilever-Beam Fracture Toughness Test for

Adhesive Joints—s s WANG 651

Analytical Modeling and ND Monitoring of Interlaminar Defects in

Fiber-Reinforced Composites—R L RAMKUMAR, S V KULKARNI, R B

PIPES, A N D S N CHATTERJEE 668

Stress Intensity Factors for a Circular Ring with Uniform Array of Radial

Cracks Using Cubic Isoparametric Singular Elements—s L PU AND

M A HUSSAIN 685

Interpretations of Crack Surface Topologies for Poly(Vinyl Chloride)—

E M SMOLEY 700

ENGINEERING APPLICATIONS

Experimental Determination of ^ | for Hollow Rectangular Tubes Containing

Corner Cracks—M E MCDERMOTT AND R I STEPHENS 719

Fracture Analysis of a Pneumatically Burst Seamless-Steel Compressed Gas

Container—B w CHRIST, J H SMITH, A N D G E HICHO 734

Crack Growth in Externally Flawed, Autofrett^ed Thick-Walled Cylinders

and Rings—J A KAPP AND R EISENSTADT 746

Estimating Fatigue Crack Propagation Lives at the Test Site—D R GALLIART 757

On the Cup and Cone Fracture of Tensile Bars—B KONG AND P C PARIS 770

SUMMARY

Summary 7^^

Index 789

STP667-EB/Aug 1979

Introduction

Developments in the field of fracture mechanics have exerted a strong influence upon the advancement of structural technology during the past decade Papers which chronicle an important part of these developments have been published in various ASTM special technical publications (STP) This volume consists of the Proceedings of the Eleventh National Symposium on Fracture Mechanics which is sponsored by ASTM Com-mittee E-24 on Fracture Testing The main body of the proceedings, consisting solely of contributed papers, presents an overview of the current state of analytical and experimental research as viewed by those members of the international technical community who participated in the Symposium A separate publication on Brittle Fracture, ASTM STP 678, consisting solely of solicited papers, has recorded the proceedings of those special sessions of the Symposium This publication delineates frontiers of research in the several areas of fracture mechanics which are addressed herein and should be of interest to scientists and engineers wishing to keep abreast of such developments

Specifically, this volume documents progress in research in several areas; the area of greatest activity being that of fatigue crack growth Papers which study the influence upon fatigue crack growth of combined fields, nonperiodic load spectra, temperature effects, crack closure and residual stresses, notches, and other effects are included Both analytical and experimental studies on stress intensity distributions and shapes of surface flaws involving finite element, boundary integral-weight function, photoelastic, and overload marking techniques are included Research on experimental techniques and the analysis of specimens is reported to-gether with new results on ATic-Jic determination and elastic-plastic fracture analysis Three papers are included which deal with the fracture

of composite materials Finally, a series of papers dealing with topics outside of the above areas which were designated as special topics are included along with a group of papers illustrating the application of fracture mechanics to problems of current and future technological importance

A feature of the Eleventh National Symposium was the announcement

by Committee E-24 Chairman J G Kaufman of the establishment of the George Rankin Irwin Medal to be awarded annually to the outstanding young researcher in the field of Fracture Mechanics The first medal was presented to Dr Irwin at the Symposium

The value of the Eleventh National Symposium on Fracture Mechanics

is evidenced by the contents of this volume and ASTM STP 678 The

contributions of the symposium organizing committee, the authors, reviewers, referees, J J Palmer and J B Wheeler of the ASTM and their staffs, together with the participation and support of P E Torgersen, Dan Frederick and J D Wilson of Virginia Polytechnic Institute and State University are gratefully acknowledged

C W Smith

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Va., 24061; symposium chairman and editor

Fatigue Crack Growth Studies

A F Liu,' / E Allison,^D F Dittmer,'

and J R Yamane^

Effect of Biaxial Stresses on

Crack Growth

REFERENCE: Liu, A F., Allison, J E., Dittmer, D F., and Yamane, J R.,

"Effect of Biaxial Stresses on Cracli Growth," Fracture Mechanics, ASTM STP

677, C W Smith, Ed., American Society for Testing and Materials, 1979, pp

5-22

ABSTRACT: This paper presents the results of a systematic evaluation of biaxial

loading on fatigue crack propagation behavior using experimental techniques, and determines the accuracy witii which a current advanced state-of-the-art analytical approach can predict this behavior These results provide data for evaluating biaxial loading effects on crack propagation

The effects of both biaxial stress ratio and applied stress level have been evaluated by conducting crack propagation tests on cruciform specimens made of 7075-T7351 and 2024-T351 aluminum alloys Tests were conducted at various

biaxial stress ratios (-1.5 £ (TJCT^ S 1.75) The magnitudes of the applied stress

were from 20 to 60 percent of the material tensile yield strength Test results indicate that biaxial stress states contribute a negligible effect on fatigue crack propagation rate as compared to data developed from uniaxial loading conditions

KEY WORDS: biaxial loading, crack propagation, fracture mechanics, metals,

structures, fatigue (materials)

Nomenclature"

a O n e half of t h e total crack length, o r the d i s t a n c e b e t w e e n a point

o n t h e A'-axis t o t h e c e n t e r of t h e cruciform s p e c i m e n , m m (in.)

Gj Projected length of " a " , p e r p e n d i c u l a r t o a^, m m (in.)

Uy Projected length of " a " , p e r p e n d i c u l a r t o ay, m m (in.)

/ Cyclic frequency ( H z )

Fty Material uniaxial tensile yield strength, k P a (ksi)

^max Stress intensity c o r r e s p o n d i n g t o a^ax, MN(M)*'^ (ksi ViiT.)

' Senior technical specialist and senior engineers, respectively, Northrop Corp., Aircraft Group, Hawthorne, Calif 90250

" Graduate student, Carnegie-Mellon University, Pittsburgh, Pa 15213, formerly Captain, United States Air Force

^min Stress intensity corresponding to o-mm, MN(M)^'^ (ksi vTiT.)

Pjc Load applied to Z-axis of a cruciform specimen, always parallel

to the crack, A^ (kip)

Py Load applied to F-axis of a cruciform specimen, always

perpen-dicular to the crack, A^ (kip)

We Width of a center cracked specimen, mm (in.)

AK K„ax - K^in, MN(M)="2 (ksi Vm".)

(Tx Stress parallel to the crack, tension or compression, kPa (ksi) (Ty Stress perpendicular to the crack, always in tension, kPa (ksi)

" Original measurements are in English units

Fracture mechanics techniques currently are being used to perform safe life analysis on aircraft and many other types of structural components Because the problems of multiaxial loading are common in aircraft airframe and engine components, it is important to evaluate and quantify multiaxial effects in order to improve the crack propagation prediction capability for design purposes

Consider that a plate, containing a through-the-thickness crack, is subjected to a biaxial stress field One of the stress components is acting perpendicular to the crack and another component parallel to the crack For problems that are solved using purely elastic formulations [7,2],^ the

crack tip stress intensity, K^, in the opening mode, theoretically is not

affected by the lateral stress component On the other hand, it can be

shown by elastic-plastic analysis, for example, \2-6], that the size of a

crack tip plastic zone varies with biaxial loading conditions Therefore,

one may speculate that the crack tip stress intensity {K) as well as the crack growth rate (da/dN) will also be influenced by the presence of the

lateral stress component Experimental data concerning the biaxial ing effect on cycUc crack growth rate, residual strength, and the direction

load-of crack growth, are available [7-14] However, the results load-of these few

investigations have been inconsistent, and none of them contained enough data to offer conclusive evidence to support, or to correlate with, the existing theories

The objective of the present investigation is to evaluate systematically the effect of the biaxial stress field on cyclic crack growth rate behavior

An analytical/experimental study program has been designed to answer the following questions:

L Do biaxial stresses affect crack tip stress intensity, cyclic crack growth rate, or crack tip plastic zone size?

2 Is an elastic - K adequate for correlating the biaxial da/dN data?

' The italic numbers in brackets refer to the list of references appended to this paper

3 Is the crack tip plastic zone important in the mechanics of constant amplitude crack growth?

The scope of the program includes the following:

1 Determination of the load-stress relationship on a selected specimen geometry

2 Determination of the elastic crack tip stress intensity factors for a crack in that geometry

3 Analytical determination of the crack tip plastic zone sizes

4 Development of experimental data and evaluation of the effects of biaxial stress ratio and applied stress level on the cyclic crack growth rate behavior of 7075-T7351 and 2024-T351 aluminum alloys

Specimen Configuration

There are many types of specimens that can be used to accomplish a

biaxial loading condition For example, Pook and Holmes [11] used a flat

cruciform specimen containing longitudinal slots in the loading arms; Beck [7] used a very large square sheet and loaded the sheet through many little straps attached around the sheet edges The criteria for designing a specimen configuration to fulfill all the objectives in the present study are:

1 The specimen should be capable of taking compression load

2 The specimen should be designed to avoid fatigue damage at the grip

or in any area other than that containing the crack

3 The size of the specimen should be large enough to minimize boundary effects on crack tip stress intensity; but it should not be too large, so that the required load levels can be kept within the capacity of the testing machine

4 The stress distribution across the specimen width should be fairly uniform

5 The specimen configuration should be simple in order to minimize machining costs

A cruciform specimen configuration has been selected for generating biaxially loaded crack growth rate data Generally, the specimen has an overall length of 597 mm (23.5 in.) including grip areas at each end of the loading arms It also has a thinner region, 6 in in diameter, in the center of the specimen An overall view of the specimen is shown in Fig 1(a)

Figure lib) is a closeup photograph of the center section Loading

conditions and dimensions of the specimen are shown in Fig 2

It has been considered that the thickness of the center region (t i) and the thickness and width of the loading arms (^2 and W) were the three

primary variables affecting the stress distribution A 17.7-mm (0.5-in.) thick loading arm was selected for ^2 to eliminate one of the three variables

(a) An overall view

(b) Closeup of the center portion (containing a curved crack)

FIG I—Cruciform specimen

l i I 9 8 16

1 I N C H - 2 ; 4 mm

\miwmiW)imMWMmi.iim^^ r^z.i—

Pv 8

1.5' i.s"—-\is.5~\-ojy- •o.iy-

1.5"—-(a) Boundary conditions

(b) Details of Area A

FIG 2—Finite element model for one quarter of a cruciform specimen

and also to minimize material and machining costs The t^ and W

dimensions have been optimized by conducting stress analysis on a dummy panel configuration (without crack) Stress distributions across the thin section were determined by using the NASTRAN computer program Figure 2 shows the finite element model representing one

quarter of the cruciform specimen Here ti =4.57 mm (0.18 in.),

W = 17.78 cm (7 in.) and (2 = 17.7 mm (0.5 in.) Also shown in Fig 2 are

three rings of triangular elements of different intermediate thicknesses (^3,

?4 and fg) to simulate the curvature connecting ti and

^2-The analytical results are presented by the curves of Figs 3 and 4 In Fig 3, the load and stress relationship at the center of the specimen is

presented; the magnitude of o-„ and a^ (per (1000 lb) ofpj are plotted as functions of Px to P^ ratio The load and stress relationship (the

NASTRAN lines in Fig 3) can be represented by:

a, = ^^^[6.55 - 1.73 (PJPJ]

cr = ^ 6 5 7 (PJP,) - 1.75]

(1) (2)

For an actual test, the required P^ and P„ values corresponding to any

desirable o^ and o-y combinations can be determined by solving Eqs 1 and

2

• ' • ROSETTE DATA POINTS

FIG 3—Load-stress relationship at the center point of a cruciform specimen

In Fig 4, Stress distributions along the Z-axis of the cruciform specimen are presented Since the specimen is symmetrical about its

center lines, the magnitudes of o^ and o-„ are plotted as functions of a The stress distributions corresponding to many Px/Py ratios were deter-

mined; however, only four typical examples are shown here

IJNCH • 2S.4inil, IKSI'ivintPl, IKIP • 4,«)N

•

; -

FIG 4—Stress gradient along the X-axis of a cruciform specimen

pair of back to back rosettes at the center of the specimen and two rosettes on each side of the center covering 69.85-mm (2.75-in.) radius, two of the side locations also had back-to-back rosettes) A guide was used to prevent out-of-plane buckling under compression loads

At each loading condition, up to four load levels were applied and two readings were taken at each load level The specimen was placed in the machine at two orientations One set of the strain gage data was taken while the F-axis of the specimen was lined up with the 244 750 N (55 kip) load cells and another set of strain gage data was taken while the F-axis of the specimen was lined up with the 342 650 N (77 kip) load cells (that is, the specimen was rotated 90 deg) Typical experimental data are plotted

on Figs 3 and 4, and it is observed in these figures that the correlations between strain gage results and the NASTRAN finite element analysis results are very good Note that in these figures, each data point represents an average of two readings Occasionally, there is a number

adjacent to a data point indicating that more than one data point was

superimposed on another; for example, the number 4 implies that the data

point represents an average of eight measurements (four load levels and

two readings per each load level) The solid and open symbols in Fig 4

indicate the results from back-to-back gages Since the results for the

back-to-back gages are almost identical, only one side of the results are

presented in Fig 3 It is significant to note that experimental data

(although not all of them are presented here) have shown that the load

response characteristics of the cruciform specimen were not affected by

the position of the specimen, that is, whether the y-axis of the specimen

was placed in line with the 342 650 N (77 kip) or the 244 750 N (55 kip)

load cells, the strain gage results were identical

Stress Intensity Factors

In the theory of linear elastic fracture mechanics, crack tip stress

intensity can be expressed as

K = a-yy/TTa-Fifl) (3)

where o-j, is the gross area stress normal to the crack In case of a center

cracked specimen (CCT), a-y will be the far field uniform stress and F(a)

accounts for the boundary conditions According to [15]

m.[.-.025(^)'.0.06(^)'].V^^™)" (4)

In case of a cruciform specimen, for a given ratio of biaxial loads, there

is a pair of stress components, (TX and a-y, at every point along a

predetermined crack plane In this case, the crack plane will be the X-axis

in Fig 2 As postulated in Ref 1, the elastic K value for a given crack

length in a biaxial state of stress should be the same as in the uniaxial

loading condition In other words, the X'-expression of Eq 3 is applicable

to the cruciform specimen except that a-y would be the (reference) stress

in the center of the uncracked specimen and F(a) would be a function of

the boundary conditions and the stress gradient of cr„ along the Z-axis

Finite element analysis of the cruciform specimen with cracks has been

conducted The finite element model of Fig 2 was used to determine

elastic K values A special "crack tip" element, originally developed by

Tong et al [16] has been incorporated into the NASTRAN In each case

analyzed, for example, each crack length, a special element was placed in

the general finite element model occupying a region representative of the

predetermined crack tip location, and the elements representing the crack

were freed from the boundary restrictions Eight specimens with crack

lengths (a = 6.35 to 69.85 mm) were loaded to various biaxial loading

ratios with (j„ = 12 ksi The results are graphed in Fig 5 Several K values calculated from Eq 3 with F{d) = 1.0 are also plotted in Fig 5 for comparison It is seen that the effects of loading conditions on elastic K

values is negligible and that the cracked cruciform specimen behaves actually like an infinite sheet especially at positive biaxial loading conditions It is even more important to note that the apparent deviations

in K, for a > 38.1 mm (1.5 in.), were mainly due to the effect of specimen

geometry rather than the effect of biaxial loading ratios The hypothesis is

substantiated by the fact that the K values for long cracks under negative

(Tx loads were actually lower than those under positive o-^ loads

Compar-ing Figs 4(a) and A{b) to Figures 4(c) and A{d) it is evident that the

tension-compression loading cases exhibited more reductions in the o-„ stresses in the area near the rim

It has been demonstrated by elastic analysis [77] that a crack will grow straight (stay on its initial path) under tension-compression biaxial stresses, but the crack will turn away from its initi£d path if the biaxial stress ratio is larger than unity, that is, if o-^ > o-j, in tension For a curved crack in a biaxial stress field, an approximate method used by Leevers et

al [12] can be used to compute X^ Since their method of analysis primarily deals with an incUned crack (with respect to either a-y or o-j.), it would be

necessary to compute both the opening mode stress intensity A" i, and the

sUding mode crack tip stress intensity, K^ Therefore, their equation has

K2 = Fiy(Ty\/lTay + FixO^xy/TTOx ( 6 )

where the factors Fiy, Fix, F^y and F^ are given in the literature (Eqs 12

through 15 of Ref 72)

When Eqs 5 and 6 are used to correlate crack growth rate or residual

strength test data, or both, it is necessary to adopt a failure criterion (or an

equivalent K value) accounting for the combined effects of K^ and K^ai

the crack tip There are numerous failure criteria available in the

litera-ture, for example Refs 18, 19, and 20 In the present study, the

following possibilities have been evaluated:

Crack Tip Plastic Zone Sizes

Using the NASTRAN computer program, elastic-plastic finite element

analyses have been conducted to determine the crack tip plastic zone

sizes in a biaxially loaded cruciform specimen Finite element models

similar to those shown in Fig 2 were used Plastic elements (not cracked)

were placed around the crack The crack tip elements were much smaller

than those used in Fig 2 having an area as small as 2.54 mm (0.1 in.) long

by 0.9525 mm (0.0375 in.) tall of an isosceles triangle, depending on the

crack size

Plastic zone sizes for five loading cases {a-x/a-y = 0, ±0.5 and ±1.0)

at (Ty = 206 700 kPa (30 ksi) for both 7075-T7351 and 2024-T351 materials,

at seven crack lengths {a = 6.35, 17.7, 25.4 and 38.1 mm for the

7075-T7351 and a = 6.35, 17.7 and 25.4 mm for the 2024-T351 specimens) have

been determined Typical results are shown in Fig 6 The dimension for rp

is the largest distance between the crack tip and the border of the plastic

zone Also in Fig 6(a) are the plastic zone sizes computed by using Eqs 9

and 31a of Ref 22; that is, rp = {KjaaxIFtyYIir and rp = IT {K^aaJFtyYI^,

respectively These theoretical values are included here to provide some

indication of the relative sizes of the crack tip plastic zone in the

cruciform specimen

(a) Plastic zone sizes for CTJ, = 0

(*) 7075-T7351,a = 1.0in

(c) 2024-T351,a=0.25in

(d) 2024-T351,a=0.5in

FIG 6—Crack tip plastic zone at 30 ksi

Facts that can be observed from all the analytic plastic zone contour maps (including those not shown in Fig 6) are listed in the following:

1 The plastic zone sizes for biaxial ratios of 0.5 and 1.0 are mately the same and are insignificantly smaller than those for the uniaxieil tension

approxi-2 The plastic zone sizes for tension-compression biaxial ratios are significantly larger than those for the uniaxial and tension-tension biaxial conditions, the higher the tension-compression ratio, the larger plastic zone size

3 Since the finite element value of rp for 7075-T7351 at a = 38.1 mm (1.5 in.) is significantly larger than the theoretical value, it seems to indicate that the crack tip plastic zone size in the cruciform specimen is a nonlinear function of the crack length

Experiments

A group of tests has been conducted to investigate the effects of biaxial

stress ratio (a-Ja-y) and applied stress level {a-ylFty) on fatigue crack

growth rate behavior of 7075-T7351 and 2024-T351 aluminum alloys Tension coupons, center cracked panels (CCT), and cruciform speci-mens (CF) were fabricated from ten sheets of 7075-T351 and four sheets of 2024-T351 plate stocks All the sheets of each material were from the same heat The size of these commercial aluminum plates was 121.92 cm (4 ft) wide by 365.76 cm (12 ft) long by 17.7 mm (0.5 in.) thick The specimens were cut from randomly selected areas- in these aluminum plates The testing conditions for the CCT specimens and the cruciform specimens are listed in Table 1 Descriptions of experimental procedures for each test type are given in the following

Tension Tests

Thirty-six tension test coupons were machined from all 14 sheets of aluminum alloys Specimen configuration was those specified in ASTM Tension Testing of Metallic Materials (E8-77a), with specimen thickness equal to 4.572 mm (0.18 in.) A 178 000 N (40 kip) MTS machine was used for conducting the tension tests

Baseline Crack Growth Rate Tests

Both the CCT and the cruciform specimens were used to develop crack growth rate data for uniaxial loading conditions (o-^ = 0) All but two of the CCT specimens were tested in a 356 000 N (80 kip) MTS machine.* The last two CCT specimens and the cruciform specimens were tested in the biaxial loading frame to ensure that compatible crack growth rate data will be developed from both the CCT and the cruciform specimens and to check out the loading characteristics of the newly built biaxial test unit through testing of the CCT specimens in both testing machines

The size of the CCT specimens was 17.78 mm (7 in.) wide by 40.64 cm (16 in.) long, having the central portion tapered down from 17.7 mm (0.5 in.) to 4.572 mm (0.18 in.) The cruciform specimen configuration has been discussed in the previous section

Since in some biaxial loading cases the crack might not grow dicular to the principal loading direction, it was desirable to measure the crack in both magnitude and direction A photographic polar grid such as that shown in Fig 1(6) was printed onto the very finely polished cruciform specimen surface Tick marks were placed at every 15 deg around the circumferential grid line, and the spacing between grid lines was 1.27 mm (0.05 in.)

perpen-* Manufactured by Material Testing Systems, Inc., Minneapolis, Minn

»o»noo>riow^'/^>/%oo>o>o'rirvirj<s i o > o & > o o & v ^ '

Both the CCT and the cruciform specimens were precracked from an initial electro discharge machining (EDM) slot, at the center of the specimen, to the desired initial flaw size (approximately 3.81 mm in total length) All specimens were precracked at test maximum load level by applying tension-tension load cycles normal to the EDM slot using the MTS machine or the biaxial loading frame, whichever is convenient Crack length measurements were made at very small increments to obtain

an adequate understanding of the crack growth behavior For the cruciform specimens, testing was terminated when the crack growth rate was faster than 2.54 //.m/cycle (10~* in./cycle) or the crack had reached the border of the flat area

The effect of cyclic frequencies was not the primary interest of this investigation However, due to the nature of the biaxial loading tests, lower frequencies had to be used for testing at higher applied loads, whereas higher frequencies could be applied to lower load test cases Therefore, as shown in Table 1, some test cases consisted of several replications, and each of them was run at a different cyclic frequency to ensure that test results would be consistent with the range of frequencies being applied

Biaxial Crack Growth Rate Tests

Fifteen 7075-T7351 specimens and fourteen 2024-T351 specimens have

been tested under various biaxial loading conditions (-1.5 ^ a-x/a-y :£ 1.75)

at various applied stress levels (0.2 s ay/Fi^ s 0.6)

A buckling guide was used in all the tension-compression biaxial ratio tests The apparatus for preventing specimen buckling consisted of two square-shaped aluminum plates and two circular steel plates The steel plates were inserted into the circular hole in the center of the aluminum plate The test specimen was sandwiched in between the aluminum plates The crack could be seen from an open slot 19.05 mm (0.75 in.) wide and 12.7 cm (5 in.) long in the center of the circular plate The circular plate could be rotated to follow the crack growth direction

The precracking and the crack growth rate recording procedures are the same as those described in the preceding paragraph

Test Results

The average tensile yield strength (the 0.2 percent offset value) for the 7075-T7351 alloy was 412 022 kPa (59.8 ksi) for both the LT and TL directions The average tensile yield strength for the 2024-T351 alloy was

367 237 kPa (53.3 ksi) for the LT direction and 319 007 kPa (46.3 ksi) for the TL direction Engineering stress-strain curves were also obtained from each tensile test One typical curve was selected from each alloy and

it was used for conducting the elastic-plastic finite element analyses

Stress intensities for cracks in the CCT specimens have been computed using Eqs 3 and 4 The crack growth rate versus A/iT plots for both aluminum alloys are presented in Fig 7

For the cruciform specimens, the test results indicated that the crack

grew straight in all the tests with (T^ ^ o-y However, when (TX exceeded

<Ty, the crack turned away from its initial plane and finally ended up propagating in a direction perpendicular to <JX (see the example shown in Fig l(i) for (Tx = 1.5 cTj,) Stress intensity values presented in Fig 5 (adjusted by the actual o-j, in each test) were used to correlate the daldN data for the straight cracks The daldN data for the curved cracks have

been analyzed by using Eqs 7 and 8 Comparing the results, it was revealed that Eq 8, which defines the effective crack tip stress intensity as

being the sum of the K values of Eqs 5 and 6, fits better with the

experimental data

Typical daldN versus ^K curves for the cruciform specimen tests are

presented in Fig 7 Examination of all the test results (including those not shown in Fig 7) has revealed that all the crack growth rate curves are almost identical; that is, for the same material and cyclic stress amplitude, there is no effect on fatigue crack growth rate due to differing biaxial

stress ratios It is significant to note that the daldN curves in Figs 7(c) and l{f), for biaxial ratios of -0.5 and +0.5, respectively, are the composites

of many sets of test data and each set of those data had been generated from different combinations of stress levels and cyclic frequencies Five test technicians were involved at different times in collecting the crack length versus cycles records for all 45 tests Even so, it is very evident that the crack growth rate behavior for all the tests has remained consistent

Summary

A series of experiments and analyses has been carried out on the cyclic crack growth behavior of center-cracked cruciform specimens under biaxial loading The results may be summarized as follows:

1 For cracks perpendicular to cr„, the effect of CTX on constant

amplitude crack growth rate is negligible

2 Cracks will grow straight except for (TX > a-y

3 Elastic K factors are obtainable for both straight and curved cracks and are adequate for correlating the biaxial daldN data

4 Analytical estimates of crack tip plastic zone size varies with biaxial ratio but experimentally this appeared to have no effect on constant amplitude crack growth rate

Acknowledgment

This work was performed under contract to the United States Air Force Flight Dynamics Laboratory, Contract F33615-76-C-3121

•

P

100

100

References

y ] Paris, P C and Sih, G C vaFracture Toughness Testing and Its Applications, ASTM

STP 381, American Society for Testing and Materials, 1965, pp 30-83

[2] Miller, K J and Kfouri, A P InternationalJournal of Fracture, Vol 10, No 3, Sept

1974, pp 393^04

[i] Kfouri, A P and Miller, K J., Paper No MS294, 4th International Conference on Fracture, Waterloo, Canada, 19-24 June 1977

\4'\ Hilton, P Ti., InternationalJournal of Fracture, Vol 9, No 2, June 1973, pp 149-156

[5] Smith, S H in Prospects of Fracture Mechanics, G C Sih, H C VanElst, and D

Broek, Eds., Noordhoff International Publishing, Leyden, The Netherlands, 1974, pp 367-388

[6] Adams, N J I., Engineering Fracture Mechanics, Vol 5, 1973, pp 983-991

[7] Beck, E J., "Fatigue Flaw Growth Behavior in Stiffened and Unstiffened Panels Loaded in Biaxial Tension," NASA Report CR-128904, National Aeronautics and Space Administration, Washington, D.C., Feb 1973

[S] Liu, A F., AIAA Journal, Vol 12, No 2, American Institute of Aeronautics and

Astronautics, Feb 1974, pp 180-185

[9] Roberts, R and Potheraj, S., Paper No L8/3, 2nd International Conference on Structural Mechanics in Reactor Technology, Beriin, Germany, 10-14 Sept 1973

[70] Kibler, J J and Roberts, R., Journal of Engineering for Industry; Transactions,

American Society of Mechanical Engineers, Series B, Nov 1970, pp lll-TiA

[Vi] Pook, L and Holmes, P., International Conference on Fatigue Testing and Design, London, England, 5-9 April 1976

[12] Leevers, P S., Radon, J C , and Culver, L E., Journal of Mechanics and Physics of

Solids, Vol 24, 1976, pp 381-395

[13] Radon, J C , Leevers, P S., and Culver, L E., Paper No MS47, 4th International

Conference on Fracture, Waterloo, Canada, 19-24 June 1977

[14] Radon, J C , Leevers, P S., and Culver, L E., Experimental Mechanics, Vol 17,

1977, pp 228-232

[15] Tada, H., Paris, P C , and Irwin, G K.,The Stress Analysis of Cracks Handbook,Del

Research Corporation, Hellertown, Pa., 1973, p 2.2

[16] Tong, P., Plan, T H H., and Lasry, S., International Journal of Numerical Mathematics in Engineering, Vol 7, 1973, pp 297-308

[17] Cotterell, B., International Journal of Fracture Mechanics, Vol 2, 1966, pp 526-533 [IS] Irwin, G R in Treatise on Adhesives and Adhesion, R L Patrick, Ed., Marcel

Dekker, New York, 1966, pp 233-267

[19] Erdogan, F and Sih, G C, Journal of Basic Engineering; Transactions, American

Society of Mechanical Engineers, Series D, Vol 85, 1963, pp 519-527

[20] Sih, G C in Mechanics of Fracture, Volume 1, Methods of Analysis and Solutions of

Crack Problems, G C Sih, Ed., Noordhoff International Pubhshing, Leyden, The Netherlands, 1973, pp XXI-XLV (Introductory Chapter)

[21] Shah, R C in Fracture Analysis, ASTM STP 560, American Society for Testing and

Materials, 1974, pp 29-52

[22] Rice, J R in Fatigue Crack Propagation, ASTM STP 415, American Society for

Testing and Materials, 1967, pp 247-311

L P Pook^ and A F Greenan^

Fatigue Crack Growth Threshold in Mild Steel Under Combined Loading

REFERENCE: Pook, L P and Greenan, A F., "Fatigue Crack Growth Threshold

in Mild Steel Under Combined Loading," Fracture Mechanics, ASTM STP 677, C

W Smith, Ed., American Society for Testing and Materials, 1979, pp 23-35

ABSTRACT: Conventional specimens used to determine fatigue crack growth

behavior have the initial crack oriented in such a way that only Mode I ments are present The fatigue-crack growth threshold behavior of mild steel in the presence of Mode II displacements was investigated by considering the fatigue behavior of spot-welded joints, where both Modes I and II are present at the point

displace-of failure, and by some experiments using specimens designed to give pure Mode II displacements

It was found that the threshold behavior is controlled by the ease with which a Mode I branch crack forms at the tip of the initial crack If such a branch forms easily, threshold behavior is controlled by AAT, for the branch crack, whereas, if branch formation is difficult, threshold behavior is controlled by AATi for the initial crack Branch crack formation seems to be facilitated by the unwanted Mode III displacements which appear when the initial crack tip is curved For the special case of nominally pure Mode II displacements, failure takes place away from the initial crack front if this is straight, and threshold behavior is determined by other factors

KEY WORDS: fatigue (materials), stress cycling, fatigue tests, mild steel, crack

propagation, crack initiation, combined loading

Nomenclature

a' Uncracked ligament (Fig 4)

flo Precrack length

E Young's modulus

K Stress intensity factor, subscripts I, II, and III denote mode Ki* Maximum value of ^ i for branch crack

AK Range o{K in fatigue cycle, subscripts, I, II, and III denote mode

AA:,C Critical value of A^i for fatigue crack growth

' Senior principal scientific officer and higher scientific officer, respectively, National Engineering Laboratory, Glasgow, Scotland

23

AX^nc Critical value of A ^ n for fatigue crack growth

P Load (Figs 4 and 7)

r Distance from crack tip

u Crack surface displacement

W Specimen width (Fig 4)

6 Angle between branch crack direction and main crack direction

cr Stress

Fatigue cracks grow perpendicular to the maximum principal applied tensile stress, or put more precisely into fracture mechanics terms, in the opening mode (Mode I, Fig 1) Like most generalizations this one has its

exceptions [1-3],^ but it does mean [7-5] that fatigue-crack growth data

can be analyzed conveniently in terms of the range of Mode I stress intensity factor, AAT] A threshold value of AK^i, AK^ic must be exceeded before a crack will grow [i,i-5]

Various techniques can be used to determine AK^ic; usually they all give essentially the same result [6] However, as with the plain specimen fatigue limit, a threshold is not necessarily defined clearly, so the precise formal definition employed can affect the numerical values obtained

National Engineering Laboratory (NEL) practice is to determine an SIN

curve for cracked specimens, with endurances plotted against the initial

values of A/sTi, and to take the threshold as the fatigue limit of this SIN

curve This method does not always give values of A^jc which are independent of initial crack size For some materials low values are obtained at very short initial crack sizes [5] However, provided the initial crack is sufficiently long, an "upper shelf value is obtained; this upper shelf value is discussed in detail in Ref J

Conventional specimens used to determine fatigue-crack growth ior have the initial crack oriented perpendicular to the applied stress A crack-like flaw from which a fatigue failure originates will not be so oriented necessarily, and crack growth, in general, will not be in the plane

behav-of the initial crack Definition behav-of threshold behavior in terms behav-of the fatigue limit of cracked specimens extends naturally to such combined mode situations; for example, A/Cnc may be defined as the critical value of AA^n, the range of the edge sliding Mode (Mode H, Fig 1) stress intensity factor,

Kii, necessary to cause crack growth which leads to failure, even though

crack growth is not in the plane of the initial crack It has been pointed out recently [7] that, for nominally pure Mode II loading, fatigue crack growth threshold behavior depends on the ease with which a Mode I branch crack forms at the tip of the initial crack This paper gives the experimental evidence on which this conclusion was based, together with the results of some further tests designed to test its validity Threshold

^ The italic numbers in brackets refer to the list of references appended to this paper

I OPENING HOOE

I EDGE SLIDING HOOE

S SHEAR MODE

FIG 1—Basic modes of crack surface displacement

behavior for combined Mode I/II loading is examined by considering the fatigue behavior of spot-welded joints, and discussed in the light of some

recently published Mode I/II threshold data for mild steel [8] Attention is

confined to cases where behavior is essentially elastic, that is, the average net section stress is less than 80 percent of the yield stress [5]

Theoretical Background

In the absence of Mode III deformations there are two plausible approaches to the estimation of combined mode threshold behavior; experimental data can be found to support either The simpler approach is

to assume that the value of A^n can have no effect on the value of A^ic on the grounds that Mode II fatigue-crack growth cannot occur [9] by the accepted mechanism for fatigue-crack growth, and also because elastic theory indicates [i] that the addition of Mode II displacement has no effect on the profile of a Mode I crack, although it does displace the profile within the cracked body This approach implies that a pure Mode II crack cannot cause fatigue failure: in other words, there is no Mode II threshold

The alternative approach is to postulate that a branch forms at the tip

of the initial (main) crack in the direction so that Ku for the branch crack has its maximum value ofKj* and ATn -*0 A number of criteria have been

suggested for the determination of this direction; a recent survey [70] pointed out the preferred direction is not well defined Some infor-mation [77] suggests that this initial direction is given approximately by

^ i sin e = A^ii (3 cos ^ - 1) (1)

26

where 6 is measured from the original crack direction Negative values of

Ki are not permitted and the root required lies in the range of ±70.5 deg

If Mode III displacements are present, the problem becomes three-dimensional and is further complicated because a preferred plane of crack growth will, in general, only intersect the initial crack front at one point

This approach implies that threshold behavior is controlled then by the

value of ^i*, so predictions require values ofKi and ^ n for the main crack

in terms ofKi* Figure 2 shows values based on numerical data for small but finite branch cracks presented graphically in Ref 12; this is inter-

preted most usefully as a failure envelope for branch crack growth under

combined Mode I/Mode II loading For pure Mode II, Ki* is about 25 percent greater than Ku for the main crack, implying that Ai^nc should be

about O.SA^ic. FOTKI = Kn, KJor the main crack is only about 55 percent ofKi*, which implies that A^ic should be reduced similarly

One problem that affects any form of combined mode testing is that, in Mode I, variations from the ideal initial crack shape lead merely to some

uncertainty in the value of Ki, whereas in the combined mode situation

such variations can introduce unwanted deformation modes For ple, in a specimen intended to give pure Mode II, crack front curvature

introduces unwanted Mode III displacements Because of the complicated three-dimensional situation, it usually is not practicable to obtain accu-rate numerical values for the stress intensity factors involved

Determination of Mode II Threshold Using Precracked Specimens

The tests were carried out on mild steel at room temperature in air The mechanical properties of the En 3 mild steel used (test mark MFLW) had

a tensile strength of 465 MN/m^ and a 0.2 percent proof stress of 330 MN/m'' The specimen design used, based on that developed by Jones and

Chisholm [13] and shown in Fig 3, is similar to that used previously [3]

but with the thickness away from the crack tips increased to improve transverse stiffness and prevent loading hole failure The sUts were spark-eroded, with a width of about 1 mm The test technique was similar

to that previously used at NEL to determine threshold data [1,3-5], and fuller details are given in Ref 14 Cracks were grown in Mode I from each

initial slit by loading in tension between the central hole and an outer hole After stress relief in vacuum for 1 h at 650°C, the precracked plates were

115

• ^

ALL DIMENSIONS IN mm

28 FRACTURE MECHANICS

tested at various load levels, with a mean to alternating load ratio of about 1.1 (the same as in Ref i ) , using the loading method shown in Fig 4 The results obtained are shown in Table 1

Values of AATn were calculated from the data [5] shown in Fig 4 Specimens failed only at the precrack having the shorter uncracked

ligament, a', and hence higher A^n- Specimens which were unbroken

after a large number of cycles were retested at a higher load; data given refer to the precrack at which failure eventually took place To ensure that conditions were essentially elastic, the average shear stress on the uncracked ligament, at maximum load, was checked and was found to be less than 80 percent of the shear yield stress, which was taken as half the 0.2 percent proof stress

The asymmetrical situation during precracking tended to cause the precrack to deviate from the desired path; deviation was not necessarily the same on both sides of a specimen, leading to a twisted precrack The two angles quoted in Table 1 are for opposite sides of the specimen, are conventionally positive when deviation is towards the outer edge, and were taken in the vicinity of the precrack tip Deviations quoted [i] for the previous tests were for an average over the precrack length The precrack fronts were all curved to some extent; the two distances shown under curvature in Table 1 are the amounts by which the precrack front trailed at the two specimen surfaces

The majority of the specimens failed at the precrack tip (tip failure); the remainder failed at the end of the spark-eroded slit (sUt failure) The specimens for the earlier tests generally had longer precracks, and many failures were caused by fretting fatigue 2 or 3 mm away from the pre-crack tip The test results for the two series are combined in Fig 5

'> u

£ £

c

0 c

c

3

II

3

FIG 5—Test results, precracked specimens

Symbols for specimens which did not fail and were retested at higher load refer to the eventual failure type Results are plotted in terms of the initial value of A^T,, for the precrack where failure took place

The results show that precrack tip failure usually requires the lowest level of AX^i, and is therefore the expected type of failure These failures define a threshold, AA^nc, of about 7.6 MN/m^'^ which is a little higher than the corresponding value [3] of A/i:,c (6.6 MN/m*"^ Other types of failure require higher load levels, and therefore only occur if for some reason tip failure is suppressed For the present tests the alternative is slit failure; not surprisingly there is considerable scatter, both because the slit tip shape was not controlled and there are variations in precrack length

For the previous similar tests \_3] the precrack length was generally much

greater, so the slit tips were in a less highly stressed region and fretting failures occurred at still higher load levels The fretting failure resuhs define an apparent threshold of about 13.6 MN/m^'^

In all cases crack growth was at an angle of roughly 70 deg to the precrack, as predicted by Eq 1, which is consistent with the view that crack growth was in Mode I, although scatter is too great for firm conclusions to be drawn The curved precrack fronts meant that crack growth was initially on a curved plane, which made it difficult to define the subsequent crack direction

A straightforward explanation would be that tip failure occurs when a Mode I branch crack forms easily However, A^nc is somewhat higher